Block Transform-Based Coding
Transform coding is a compression technique used in many audio, image and video compression systems. Uncompressed digital image and video is typically represented or captured as samples of picture elements or colors at locations in an image or video frame arranged in a two-dimensional (2D) grid. This is referred to as a spatial-domain representation of the image or video. For example, a typical format for images consists of a stream of 24-bit color picture element samples arranged as a grid. Each sample is a number representing color components at a pixel location in the grid within a color space, such as RGB, or YIQ, among others. Various image and video systems may use various different color, spatial and time resolutions of sampling. Similarly, digital audio is typically represented as time-sampled audio signal stream. For example, a typical audio format consists of a stream of 16-bit amplitude samples of an audio signal taken at regular time intervals.
Uncompressed digital audio, image and video signals can consume considerable storage and transmission capacity. Transform coding reduces the size of digital audio, images and video by transforming the spatial-domain representation of the signal into a frequency-domain (or other like transform domain) representation, and then reducing resolution of certain generally less perceptible frequency components of the transform-domain representation. This generally produces much less perceptible degradation of the digital signal compared to reducing color or spatial resolution of images or video in the spatial domain, or of audio in the time domain.
More specifically, a typical block transform-based codec 100 shown in FIG. 1 divides the uncompressed digital image's pixels into fixed-size two dimensional blocks (X1, . . . Xn), each block possibly overlapping with other blocks. A linear transform 120-121 that does spatial-frequency analysis is applied to each block, which converts the spaced samples within the block to a set of frequency (or transform) coefficients generally representing the strength of the digital signal in corresponding frequency bands over the block interval. For compression, the transform coefficients may be selectively quantized 130 (i.e., reduced in resolution, such as by dropping least significant bits of the coefficient values or otherwise mapping values in a higher resolution number set to a lower resolution), and also entropy or variable-length coded 130 into a compressed data stream. At decoding, the transform coefficients will inversely transform 170-171 to nearly reconstruct the original color/spatial sampled image/video signal (reconstructed blocks {circumflex over (X)}1, . . . {circumflex over (X)}n).
The block transform 120-121 can be defined as a mathematical operation on a vector x of size N. Most often, the operation is a linear multiplication, producing the transform domain output y=M x, M being the transform matrix. When the input data is arbitrarily long, it is segmented into N sized vectors and a block transform is applied to each segment. For the purpose of data compression, reversible block transforms are chosen. In other words, the matrix M is invertible. In multiple dimensions (e.g., for image and video), block transforms are typically implemented as separable operations. The matrix multiplication is applied separably along each dimension of the data (i.e., both rows and columns).
For compression, the transform coefficients (components of vector y) may be selectively quantized (i.e., reduced in resolution, such as by dropping least significant bits of the coefficient values or otherwise mapping values in a higher resolution number set to a lower resolution), and also entropy or variable-length coded into a compressed data stream.
At decoding in the decoder 150, the inverse of these operations (dequantization/entropy decoding 160 and inverse block transform 170-171) are applied on the decoder 150 side, as show in FIG. 1. While reconstructing the data, the inverse matrix M−1 (inverse transform 170-171) is applied as a multiplier to the transform domain data. When applied to the transform domain data, the inverse transform nearly reconstructs the original time-domain or spatial-domain digital media.
In many block transform-based coding applications, the transform is desirably reversible to support both lossy and lossless compression depending on the quantization factor. With no quantization (generally represented as a quantization factor of 1) for example, a codec utilizing a reversible transform can exactly reproduce the input data at decoding. However, the requirement of reversibility in these applications constrains the choice of transforms upon which the codec can be designed.
Many image and video compression systems, such as MPEG and Windows Media, among others, utilize transforms based on the Discrete Cosine Transform (DCT). The DCT is known to have favorable energy compaction properties that result in near-optimal data compression. In these compression systems, the inverse DCT (IDCT) is employed in the reconstruction loops in both the encoder and the decoder of the compression system for reconstructing individual image blocks.
Entropy Coding of Wide-Range Transform Coefficients
Wide dynamic range input data leads to even wider dynamic range transform coefficients generated during the process of encoding an image. For instance, the transform coefficients generated by an N×N DCT operation have a dynamic range greater than N times the dynamic range of the original data. With small or unity quantization factors (used to realize low-loss or lossless compression), the range of quantized transform coefficients is also large. Statistically, these coefficients have a Laplacian distribution as shown in FIGS. 2 and 3. FIG. 2 shows a Laplacian distribution for wide dynamic range coefficients. FIG. 3 shows a Laplacian distribution for typical narrow dynamic range coefficients.
Conventional transform coding is tuned for a small dynamic range of input data (typically 8 bits), and relatively large quantizers (such as numeric values of 4 and above). FIG. 3 is therefore representative of the distribution of transform coefficients in such conventional transform coding. Further, the entropy encoding employed with such conventional transform coding can be a variant of run-level encoding, where a succession of zeroes is encoded together with a non-zero symbol. This can be an effective means to represent runs of zeroes (which occur with high probability), as well as capturing inter-symbol correlations.
On the other hand, conventional transform coding is less suited to compressing wide dynamic range distributions such as that shown in FIG. 2. Although the symbols are zero with higher probability than any other value (i.e., the distribution peaks at zero), the probability of a coefficient being exactly zero is miniscule for the wide dynamic range distribution. Consequently, zeroes do not occur frequently, and run length entropy coding techniques that are based on the number of zeroes between successive non-zero coefficients are highly inefficient for wide dynamic range input data.
The wide dynamic range distribution also has an increased alphabet of symbols, as compared to the narrow range distribution. Due to this increased symbol alphabet, the entropy table(s) used to encode the symbols will need to be large. Otherwise, many of the symbols will end up being escape coded, which is inefficient. The larger tables require more memory and may also result in higher complexity.
The conventional transform coding therefore lacks versatility—working well for input data with the narrow dynamic range distribution, but not on the wide dynamic range distribution.